Maximal among abelian subgroups of maximum rank

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This article is about a maximality notion among subgroups, related to abelianness or small class, in a group of prime power order.
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Definition

A subgroup of a group of prime power order is termed maximal among abelian subgroups of maximum rank if it satisfies the following equivalent conditions:

1. It is an abelian subgroup of maximum rank and is not contained in any bigger abelian subgroup of maximum rank.
2. It is both an abelian subgroup of maximum rank and maximal among abelian subgroups.

Equivalence of definitions

Note that since the rank is monotone (i.e., the rank of a subgroup is at most equal to the rank of the whole group), any abelian subgroup contained an abelian subgroup of maximum rank also has maximum rank. This explains the equivalence of the definitions.